Even though dividends are discrete, literature often makes the assumption of continuous dividends (mostly in the case of indices but the individual stocks as well).
The dividend yield denoted by q is often considered as an adjustment to the risk free rate (i.e. r-q).
My question is, what happens to American Call options if r-q < 0? Is it now possible to exercise before maturity so it can no longer be calculated as a European option? Logic says you can early exercise but I am not sure.
Some footnote: In discrete dividend case we know that we should only exercise American Calls before maturity if the excess value of the option is less than the dividend. Otherwise value of the American Option will always be greater than the exercise price. This is mainly due to r > 0, and in the rare case of r < 0 American Puts become equivalent to European Puts.